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Mirrors > Home > ILE Home > Th. List > nfald | GIF version |
Description: If 𝑥 is not free in 𝜑, it is not free in ∀𝑦𝜑. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 6-Jan-2018.) |
Ref | Expression |
---|---|
nfald.1 | ⊢ Ⅎ𝑦𝜑 |
nfald.2 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
Ref | Expression |
---|---|
nfald | ⊢ (𝜑 → Ⅎ𝑥∀𝑦𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfald.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | nfri 1499 | . . 3 ⊢ (𝜑 → ∀𝑦𝜑) |
3 | nfald.2 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
4 | 2, 3 | alrimih 1449 | . 2 ⊢ (𝜑 → ∀𝑦Ⅎ𝑥𝜓) |
5 | nfnf1 1524 | . . . 4 ⊢ Ⅎ𝑥Ⅎ𝑥𝜓 | |
6 | 5 | nfal 1556 | . . 3 ⊢ Ⅎ𝑥∀𝑦Ⅎ𝑥𝜓 |
7 | hba1 1520 | . . . 4 ⊢ (∀𝑦Ⅎ𝑥𝜓 → ∀𝑦∀𝑦Ⅎ𝑥𝜓) | |
8 | sp 1491 | . . . . 5 ⊢ (∀𝑦Ⅎ𝑥𝜓 → Ⅎ𝑥𝜓) | |
9 | 8 | nfrd 1500 | . . . 4 ⊢ (∀𝑦Ⅎ𝑥𝜓 → (𝜓 → ∀𝑥𝜓)) |
10 | 7, 9 | hbald 1471 | . . 3 ⊢ (∀𝑦Ⅎ𝑥𝜓 → (∀𝑦𝜓 → ∀𝑥∀𝑦𝜓)) |
11 | 6, 10 | nfd 1503 | . 2 ⊢ (∀𝑦Ⅎ𝑥𝜓 → Ⅎ𝑥∀𝑦𝜓) |
12 | 4, 11 | syl 14 | 1 ⊢ (𝜑 → Ⅎ𝑥∀𝑦𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1333 Ⅎwnf 1440 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1427 ax-7 1428 ax-gen 1429 ax-4 1490 ax-ial 1514 |
This theorem depends on definitions: df-bi 116 df-nf 1441 |
This theorem is referenced by: dvelimALT 1990 dvelimfv 1991 nfeudv 2021 nfeqd 2314 nfraldw 2489 nfraldxy 2490 nfiotadw 5141 nfixpxy 6665 bdsepnft 13559 |
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